villain is a solid-looking reg: 21/16/3 after 483 hands.
I was experimenting with checking the turn against this opponent since I assume he is pretty decent and isn't likely to call 2 barrels with the worse hand, whereas he might float me or bet if he senses weakness.

This kind of backfired when my check raise was met with a shove.

do I need to fold once villain shoves? I thought maybe, because I had underrepresented my hand ( is that true?) He might just think I was full of it could be shoving with the same hand as me or even KQ, but perhaps I'm being wildly overoptimistic?

I like this line and I ahve pretty much the same thought process as u do
3 times I did it on NL50, once reg folded, once he showed and once he called, when he showed he had a set and call was w JJ or smth like that (I had AKo too
)

kind of strange to c/r AK for value and then fold to shove
So once you c/r I would prefer calling shove since there are two flushdraws and he can easily shove them
Also I think very fine line is just to c/c turn and river or b/f turn and c/c river
Then you can go to showdown and see what kind of hands he is playing when he is c/c on K72ss board